Conjugacy problem in groups of non-oriented geometrizable 3-manifolds

The conjugacy problem in groups of orientable geometrizable 3-manifolds

We prove that fundamental groups of orientable (geometrizable) 3-manifolds have a solvable conjugacy problem.

The conjugacy problem in groups of non-orientable 3-manifolds

We prove that fundamental groups of non-orientable 3-manifolds have a solvable conjugacy problem, and construct an algorithm. Together with our earlier work on the conjugacy problem in groups on orientable geometrizable 3-manifolds, all $π_1$ of (geometrizable) 3-manifolds have a solvable conjugacy problem. In corollaries, both the twisted conjugacy problem in closed surface groups and the conjugacy problem in closed surface-by-cyclic groups, are solvable.

通过覆盖的同调群刻画 Riemann 曲面上的闭曲线

Characterizing closed curves on Riemann surfaces via homology groups of coverings

本文利用无分歧 $p$-覆盖紧化同调群中的辛子模检测闭曲线几何交，并通过 $p$-进 Reidemeister 配对刻画正交性，进而几何证明有向曲面群的共轭 $p$-可分性。

从带一个标记元的有限商中识别定向纽结

Recognizing oriented knots from finite quotients with one marked element

该研究证明，由μ_Kλ_K共轭类标记的纽结群的profinite完备化可确定定向纽结的等价类，其方法结合了Xu的真性定理等，适用于卫星与复合纽结，且有限群上的Eisermann染色多项式可确定定向纽结。

Tadpole Labelled Oriented Graph Groups and Cyclically Presented Groups

We study a class of Labelled Oriented Graph (LOG) group where the underlying graph is a tadpole graph. We show that such a group is the natural HNN extension of a cyclically presented group and investigate the relationship between the LOG group and the cyclically presented group. We relate the second homotopy groups of their presentations and show that hyperbolicity of the cyclically presented group implies solvability of the conjugacy problem for the LOG group. In the case where the label on the tail of the LOG spells a positive word in the vertices in the circuit we show that the LOGs and gr…

Residual $p$ properties of mapping class groups and surface groups

Let $\mathcal M (Σ, \mathcal P)$ be the mapping class group of a punctured oriented surface $(Σ, \mathcal P)$ (where $\mathcal P$ may be empty), and let $\mathcal T_p(Σ,\mathcal P)$ be the kernel of the action of $\mathcal M (Σ, \mathcal P)$ on $H_1 (Σ\setminus \mathcal P, \mathbb F_p)$. We prove that $\mathcal T_p(Σ, \mathcal P)$ is residually $p$. In particular, this shows that $\mathcal M (Σ, \mathcal P)$ is virtually residually $p$. For a group $G$ we denote by $\mathcal I_p(G)$ the kernel of the natural action of ${\rm Out} (G)$ on $H_1(G, \mathbb F_p)$. In order to achieve our theorem, w…